def 			lowercase(	UpperCamelCase_							,					UpperCamelCase_       )		->	Optional[int]:







 '''simple docstring'''
 return (pointa[0] - pointa[0]) ** 2 + (pointa[1] - pointa[1]) ** 2





def 			lowercase(	UpperCamelCase_							,					UpperCamelCase_=0       )		->	Any:







 '''simple docstring'''
 return sorted(UpperCamelCase_							,					key=lambda UpperCamelCase_       : x[column]       )





def 			lowercase(	UpperCamelCase_							,					UpperCamelCase_							,					UpperCamelCase_=float("""inf"""       )       )		->	Optional[Any]:







 '''simple docstring'''

 for i in range(points_counts - 1       ):
  for j in range(i + 1							,					UpperCamelCase_       ):
   UpperCamelCase			  =      euclidean_distance_sqr(points[i]							,					points[j]       )
   if current_dis < min_dis:
    UpperCamelCase			  =      current_dis
 return min_dis





def 			lowercase(	UpperCamelCase_							,					UpperCamelCase_							,					UpperCamelCase_=float("""inf"""       )       )		->	str:







 '''simple docstring'''

 for i in range(min(6							,					points_counts - 1       )							,					UpperCamelCase_       ):
  for j in range(max(0							,					i - 6       )							,					UpperCamelCase_       ):
   UpperCamelCase			  =      euclidean_distance_sqr(points[i]							,					points[j]       )
   if current_dis < min_dis:
    UpperCamelCase			  =      current_dis
 return min_dis





def 			lowercase(	UpperCamelCase_							,					UpperCamelCase_							,					UpperCamelCase_       )		->	str:







 '''simple docstring'''

 # base case
 if points_counts <= 3:
  return dis_between_closest_pair(UpperCamelCase_							,					UpperCamelCase_       )

 # recursion
 UpperCamelCase			  =      points_counts // 2
 UpperCamelCase			  =      closest_pair_of_points_sqr(
     UpperCamelCase_							,					points_sorted_on_y[:mid]							,					UpperCamelCase_       )
 UpperCamelCase			  =      closest_pair_of_points_sqr(
     UpperCamelCase_							,					points_sorted_on_y[mid:]							,					points_counts - mid       )
 UpperCamelCase			  =      min(UpperCamelCase_							,					UpperCamelCase_       )
 UpperCamelCase			  =      []
 for point in points_sorted_on_x:
  if abs(point[0] - points_sorted_on_x[mid][0]       ) < closest_pair_dis:
   cross_strip.append(UpperCamelCase_       )

 UpperCamelCase			  =      dis_between_closest_in_strip(
     UpperCamelCase_							,					len(UpperCamelCase_       )							,					UpperCamelCase_       )
 return min(UpperCamelCase_							,					UpperCamelCase_       )





def 			lowercase(	UpperCamelCase_							,					UpperCamelCase_       )		->	Optional[int]:







 '''simple docstring'''
 UpperCamelCase			  =      column_based_sort(UpperCamelCase_							,					column=0       )
 UpperCamelCase			  =      column_based_sort(UpperCamelCase_							,					column=1       )
 return (
     closest_pair_of_points_sqr(
         UpperCamelCase_							,					UpperCamelCase_							,					UpperCamelCase_       )
 ) ** 0.5


if __name__ == "__main__":
     _SCREAMING_SNAKE_CASE 		=      [(2, 3), (1_2, 3_0), (4_0, 5_0), (5, 1), (1_2, 1_0), (3, 4)]
     print("""Distance:""", closest_pair_of_points(points, len(points)))

def 			lowercase(	UpperCamelCase_ = 100       )		->	int:







 '''simple docstring'''
 UpperCamelCase			  =      n * (n + 1) * (2 * n + 1) / 6
 UpperCamelCase			  =      (n * (n + 1) / 2) ** 2
 return int(square_of_sum - sum_of_squares       )


if __name__ == "__main__":
     print(F'''{solution() = }''')
